Symplectic Induction, Prequantum Induction, and Prequantum Multiplicities

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary G-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian G-spaces, which (as we show) unfortunately fails to mirror the situation where more than one G-module "quantizes" a given Hamiltonian G-space. This paper offers evidence that the situation is remedied by working in the category of *prequantum* G-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces, and establish Frobenius reciprocity as well as the "induction in stages" property.